On a Smooth Dual Gap Function for a Class of Quasi-Variational Inequalities

A well-known technique for the solution of quasi-variational inequalities (QVIs) consists in the reformulation of QVIs as a constrained or unconstrained optimization problem by means of so-called gap functions. In contrast to standard variational inequalities, however, these gap functions turn out to be nonsmooth in general. Here it is shown that one can obtain an unconstrained optimization reformulation of a class of QVIs by using a continuously differentiable dual gap function. This extends an idea from Dietrich (Journal of Mathematical Analysis and Applications 235, 1999, pp. 380–393). Some numerical results illustrate the practical behavior of this dual gap function approach.